Level 0: No Automation
Extremely simple. The car is not automated in any way and relies exclusively on a human for all the tasks.
Level 1: Driver Assitance
This level still involves both the human and the computer system cooperating together on tasks.
While the assistance systems take action on features like steering, acceleration/deceleration (as well as the ones that involve "information about the driving environment") the human is expected to do the majority of the work.
Level 2: Partial Automation
Here's where things start to get more serious.
In level 2 systems, the steering and speed of the vehicle are controlled by "one or more driver assistance systems" but human controls the other elements of driving.
Level 3: Partial Automation
Vehicles in level 3 and above are considered "automated driving systems". The main difference here is that the vehicles are now able to monitor the driving environment around them and then make decisions themselves. For instance, a level 3 car is capable of seeing a slower moving vehicle in front of it and overtake. The driver though is still there if things go wrong.
Level 4: High Automation
SAE describes this as "having a driving mode-specific performance by an automated driving system of all aspects of the dynamic driving task".
In simple words, the human inside the cockpit doesn't really need to pay attention to the road cause if things go wrong the system can handle it itself. But will probably still remain seated in the traditional driving position.
Level 5: Full Automation
In this scenario, the vehicle has a full level of automation and therefore don't need any pedals, steering wheels, or controls for a human to take charge.
No human control of a vehicle is needed at all.